The formula for the univariate derivative is as follows:
f ′ ( x ) = lim Δ x → 0 f ( x + Δ x ) − f ( x ) Δ x ( 1 ) f^{'}(x)=\lim \limits_{ \Delta x \to 0} \frac{f(x+\Delta x)-f(x)}{\Delta x} \qquad (1)f′(x)=Δx→0limΔx _f(x+Δ x )−f(x)( 1 )
Formula (1) is equivalent to
f ′ ( x ) ≈ f ( x + Δ x ) − f ( x ) Δ x Δ x takes a very small value f^{'}(x) \approx \frac{ f(x+\Delta x)-f(x)}{\Delta x} \qquad \Delta x takes a very small valuef′(x)≈Δx _f(x+Δ x )−f(x)Δ x takes a very small value ,
that is,f ( x + Δ x ) − f ( x ) ≈ f ′ ( x ) ⋅ Δ xf(x+\Delta x)-f(x) \approx f^{'}(x) ·\Delta xf(x+Δ x )−f(x)≈f′(x)⋅Δx _
extended toApproximate formulas for multivariate functions
f ( x + Δ x , y + Δ y ) − f ( x , y ) ≈ ∂ f ∂ x ⋅ Δ x + ∂ f ∂ y ⋅ Δ yf(x+\Delta x,y+\Delta y)-f(x ,y) \approx \frac{\partial f}{\partial x} ·\Delta x+\frac{\partial f}{\partial y} ·\Delta yf(x+Δx , _y+y ) _−f(x,y)≈∂x∂f⋅Δx _+∂y∂f⋅Δy
令 Δ z = f ( x + Δ x , y + Δ y ) − f ( x , y ) \Delta z=f(x+\Delta x,y+\Delta y)-f(x,y) Δz=f(x+Δx , _y+y ) _−f(x,y )，即Δ z = ∂ f ∂ x ⋅ Δ x + ∂ f ∂ y ⋅ Δ y \Delta z=\frac{\partial f}{\partial x} ·\Delta x+\frac{\partial f}{ \partial y} ·\Delta yΔz=∂x∂f⋅Δx _+∂y∂f⋅Δ y
expressed as an inner product is:
Δ z = ( ∂ f ∂ x , ∂ f ∂ y ) . ( Δ x , Δ y ) \Delta z=(\frac{\partial f}{\partial x} ,\frac{\partial f}{\partial y} ).(\Delta x,\Delta y)Δz=(∂x∂f,∂y∂f) . ( Δ x ,y ) _